The computations performed in the routine
noetherNormalization use a random linear change of coordinates, hence one should expect the output to change each time the routine is executed.
i1 : R = QQ[x_1..x_4];
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i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o2 : Ideal of R
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i3 : (f,J,X) = noetherNormalization I
7 3 2 13 2 3
o3 = (map(R,R,{-x + -x + x , x , 5x + -x + x , x }), ideal (--x + -x x
6 1 8 2 4 1 1 3 2 3 2 6 1 8 1 2
------------------------------------------------------------------------
35 3 191 2 2 1 3 7 2 3 2 2
+ x x + 1, --x x + ---x x + -x x + -x x x + -x x x + 5x x x +
1 4 6 1 2 72 1 2 4 1 2 6 1 2 3 8 1 2 3 1 2 4
------------------------------------------------------------------------
2 2
-x x x + x x x x + 1), {x , x })
3 1 2 4 1 2 3 4 4 3
o3 : Sequence
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The next example shows how when we use the lexicographical ordering, we can see the integrality of
R/ f I over the polynomial ring in
dim(R/I) variables:
i4 : R = QQ[x_1..x_5, MonomialOrder => Lex];
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i5 : I = ideal(x_2*x_1-x_5^3, x_5*x_1^3);
o5 : Ideal of R
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i6 : (f,J,X) = noetherNormalization I
3 2 1
o6 = (map(R,R,{-x + 4x + x , x , -x + 3x + x , -x + 2x + x , x }),
2 1 2 5 1 3 1 2 4 2 1 2 3 2
------------------------------------------------------------------------
3 2 3 27 3 2 2 27 2 3
ideal (-x + 4x x + x x - x , --x x + 27x x + --x x x + 72x x +
2 1 1 2 1 5 2 8 1 2 1 2 4 1 2 5 1 2
------------------------------------------------------------------------
2 9 2 4 3 2 2 3
36x x x + -x x x + 64x + 48x x + 12x x + x x ), {x , x , x })
1 2 5 2 1 2 5 2 2 5 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 6x_1x_2x_5^6-864x_2^9x_5-6144x_2^9+108x_2^8x_5^2+1536x_2^8x_5-9x
{-9} | 3072x_1x_2^2x_5^3-54x_1x_2x_5^5+768x_1x_2x_5^4+7776x_2^9-972x_2^
{-9} | 100663296x_1x_2^3+1769472x_1x_2^2x_5^2+50331648x_1x_2^2x_5+1458x
{-3} | 3x_1^2+8x_1x_2+2x_1x_5-2x_2^3
------------------------------------------------------------------------
_2^7x_5^3-384x_2^7x_5^2+96x_2^6x_5^3-24x_2^5x_5^4+6x_2^4x_5^5+16x_2^2x_5
8x_5-4608x_2^8+81x_2^7x_5^2+2304x_2^7x_5-864x_2^6x_5^2+216x_2^5x_5^3-54x
_1x_2x_5^5-10368x_1x_2x_5^4+294912x_1x_2x_5^3+6291456x_1x_2x_5^2-209952x
------------------------------------------------------------------------
^6+4x_2x_5^7
_2^4x_5^4+768x_2^4x_5^3+8192x_2^3x_5^3-144x_2^2x_5^5+4096x_2^2x_5^4-36x_
_2^9+26244x_2^8x_5+186624x_2^8-2187x_2^7x_5^2-77760x_2^7x_5+221184x_2^7+
------------------------------------------------------------------------
2x_5^6+512x_2x_5^5
23328x_2^6x_5^2-165888x_2^6x_5-2359296x_2^6-5832x_2^5x_5^3+41472x_2^5x_5
------------------------------------------------------------------------
^2+589824x_2^5x_5+25165824x_2^5+1458x_2^4x_5^4-10368x_2^4x_5^3+294912x_2
------------------------------------------------------------------------
^4x_5^2+6291456x_2^4x_5+268435456x_2^4+4718592x_2^3x_5^2+201326592x_2^3x
------------------------------------------------------------------------
_5+3888x_2^2x_5^5-27648x_2^2x_5^4+1966080x_2^2x_5^3+50331648x_2^2x_5^2+
------------------------------------------------------------------------
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972x_2x_5^6-6912x_2x_5^5+196608x_2x_5^4+4194304x_2x_5^3 |
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5 1
o7 : Matrix R <--- R
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If
noetherNormalization is unable to place the ideal into the desired position after a few tries, the following warning is given:
i8 : R = ZZ/2[a,b];
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i9 : I = ideal(a^2*b+a*b^2+1);
o9 : Ideal of R
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i10 : (f,J,X) = noetherNormalization I
--warning: no good linear transformation found by noetherNormalization
2 2
o10 = (map(R,R,{a + b, a}), ideal(a b + a*b + 1), {b})
o10 : Sequence
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Here is an example with the option
Verbose => true:
i11 : R = QQ[x_1..x_4];
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i12 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o12 : Ideal of R
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i13 : (f,J,X) = noetherNormalization(I,Verbose => true)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20
1 7 2
o13 = (map(R,R,{5x + x + x , x , -x + -x + x , x }), ideal (6x + x x +
1 2 4 1 2 1 3 2 3 2 1 1 2
-----------------------------------------------------------------------
5 3 73 2 2 7 3 2 2 1 2 7 2
x x + 1, -x x + --x x + -x x + 5x x x + x x x + -x x x + -x x x
1 4 2 1 2 6 1 2 3 1 2 1 2 3 1 2 3 2 1 2 4 3 1 2 4
-----------------------------------------------------------------------
+ x x x x + 1), {x , x })
1 2 3 4 4 3
o13 : Sequence
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The first number in the output above gives the number of linear transformations performed by the routine while attempting to place
I into the desired position. The second number tells which
BasisElementLimit was used when computing the (partial) Groebner basis. By default,
noetherNormalization tries to use a partial Groebner basis. It does this by sequentially computing a Groebner basis with the option
BasisElementLimit set to predetermined values. The default values come from the following list:
{5,20,40,60,80,infinity}. To set the values manually, use the option
LimitList:
i14 : R = QQ[x_1..x_4];
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i15 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o15 : Ideal of R
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i16 : (f,J,X) = noetherNormalization(I,Verbose => true,LimitList => {5,10})
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 10
1 1 3 10 3 2 1
o16 = (map(R,R,{-x + -x + x , x , -x + --x + x , x }), ideal (-x + -x x
2 1 2 2 4 1 5 1 3 2 3 2 2 1 2 1 2
-----------------------------------------------------------------------
3 3 59 2 2 5 3 1 2 1 2 3 2
+ x x + 1, --x x + --x x + -x x + -x x x + -x x x + -x x x +
1 4 10 1 2 30 1 2 3 1 2 2 1 2 3 2 1 2 3 5 1 2 4
-----------------------------------------------------------------------
10 2
--x x x + x x x x + 1), {x , x })
3 1 2 4 1 2 3 4 4 3
o16 : Sequence
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To limit the randomness of the coefficients, use the option
RandomRange. Here is an example where the coefficients of the linear transformation are random integers from
-2 to
2:
i17 : R = QQ[x_1..x_4];
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i18 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o18 : Ideal of R
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i19 : (f,J,X) = noetherNormalization(I,Verbose => true,RandomRange => 2)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20
--trying with basis element limit: 40
--trying with basis element limit: 60
--trying with basis element limit: 80
--trying with basis element limit: infinity
--trying random transformation: 2
--trying with basis element limit: 5
--trying with basis element limit: 20
2
o19 = (map(R,R,{- 2x + x , x , 2x - 3x + x , x }), ideal (x - 2x x +
2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
2 2 3 2 2 2
x x + 1, - 4x x + 6x x - 2x x x + 2x x x - 3x x x + x x x x +
1 4 1 2 1 2 1 2 3 1 2 4 1 2 4 1 2 3 4
-----------------------------------------------------------------------
1), {x , x })
4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.